They ride a bus
They are riding on a bus should be also accepted
For transportation in Ukrainian language the correct is to use instrumental case Вони їдуть автобусом ( автобусом means by car) ( на автобусі means "in the car")
Автобус means "bus," not "car."
Why is THE bus not accepted
Accepted now for sure! (don't know since when)
Why is they are going on a bus not accepted?
It is. You probably made a typo or some other mistake.
For example, I see a report submitted by someone asking to accept "The are going on a bus". Could be you? :)